Question

Complete the proof of the Law of Sines/Cosines.

Given triangle ABC with altitude segment BD labeled x. Angles ADB and CDB are right angles by _____1._____, making triangle ABD and triangle BCD right triangles. Using the trigonometric ratios sine of A equals x over c and sine of C equals x over a. Multiplying to isolate x in both equations gives x = _____2._____ and x = a ⋅ sinC. We also know that x = x by the reflexive property. By the substitution property, _____3._____. Dividing each side of the equation by ac gives: sine of A over a equals sine of C over c.


1. definition of altitude

2. c ⋅ sinA

3. c ⋅ sinA = a ⋅ sinC

1. definition of right triangles

2. c ⋅ sinB

3. c ⋅ sinB = a ⋅ sinC

1. definition of right triangles

2. a ⋅ sinA

3. a ⋅ sinA = c ⋅ sinC

1. definition of altitude

2. c ⋅ sinA

3. a ⋅ sinA = c ⋅ sinC

Answer 1

Answer: answer A

Explanation:

1. Definition of altitude

2. c • sinA

3. c • SinA= a • sinC

Answer 2

(A)

1. Definition of altitude
2. c⋅sinA
3. 
   `c\cdot sinA=a\cdot sinC`.

Explanation:

Law of Sines/Cosines.

• Given triangle ABC with altitude segment BD labeled x. Angles ADB and CDB are right angles by the (1) definition of altitude making triangle ABD and triangle BCD right triangles.

• Using the trigonometric ratios
  `sin A=(x)/(c)` and
  `sin C=(x)/(a)`.

• Multiplying to isolate x in both equations gives (2)
  `x=c\cdot sinA` and
  `x=a\cdot sinC.`

• We also know that x = x by the reflexive property.

• By the substitution property,(3)
  `c\cdot sinA=a\cdot sinC`.

• Dividing each side of the equation by ac gives:

`(c\cdot sinA)/(ac) =(a\cdot sinC)/(ac) \\( sinA)/(a) =(sinC)/(c)`